BEVL Toolkit — Break-Even Volatility Surface Construction

Break-Even Volatility (BEVL) is the volatility under which the expected P&L of a delta-hedged option is zero (Dupire, 2006).
Unlike implied volatility (IV), which comes from option market prices, BEVL is constructed from historical paths with Gamma weighting.
This provides a “virtual volatility surface” that serves as a fair benchmark for skew and risk premium analysis.

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Features

• Build BEVL surfaces from historical paths or simulations
• Multiple weighting schemes:
  • Equal weights
  • Realized-vol similarity weights (Gaussian kernel)
  • KL-implied weights (Weighted Monte Carlo calibration)
• Gamma-weighted integration across paths and strikes
• Brent root-finding for BEVL at each strike/tenor
• Example: S&P500 1M BEVL surface

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Methodology

1. Continuous Definition

Break-even volatility $\sigma_{BE}$ is defined as the volatility under which the expected P&L of a delta-hedged option equals zero:

$$\mathbb{E}\!\left[ \int_{0}^{T} e^{-rt}\, S_t^2\, \Gamma_{\mathrm{BS}}(t,S_t;\sigma_{\mathrm{BE}}) \big(\sigma_t^2 - \sigma_{\mathrm{BE}}^2\big)\, dt \right] = 0.$$

where:

• $\Gamma_{\text{BS}}$ = Black–Scholes Gamma under constant volatility assumption
• $\sigma_t^2$ = realized variance at time $t$

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2. Discretization

With $N$ historical/simulated paths ($n$) and $K$ time steps ($k$):

$$ \sigma_{n,k}^2 \approx \frac{\big(\ln S_{n,k+1}-\ln S_{n,k}\big)^2}{\Delta t_k} $$

where:

• $S_{n,k}$ = price on path $n$ at step $k$

• Realized variance estimate: $$ \sigma_{n,k}^2 \approx \frac{\big(\ln S_{n,k+1}-\ln S_{n,k}\big)^2}{\Delta t_k} $$

• $p_n$ = weight of path $n$

• Solve via root-finding in $\sigma$

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3. Black–Scholes Dollar Gamma

For $\tau = T - t$:

$$ S^2 \Gamma_{\text{BS}}(t, S; \sigma) ;=; e^{-q\tau} , \frac{S , \phi(d_1)}{\sigma \sqrt{\tau}}, $$

where:

• $d_1 = \dfrac{\ln(S/K) + (r - q + \tfrac{1}{2}\sigma^2)\tau}{\sigma \sqrt{\tau}}$
• $\phi(\cdot)$ = standard normal PDF

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4. Path Weights

We consider different schemes for weighting paths:

(a) Equal Weights

$$ p_n = \frac{1}{N} $$

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(b) Realized-Vol Similarity Weights

Compare each path’s realized volatility $\hat{\sigma}n$ to a reference volatility $\sigma{\text{ref}}$:

$$ \tilde{w}_n = \exp!\left(-\frac{(\hat{\sigma}_n - \sigma_{\text{ref}})^2}{2h^2}\right), \qquad p_n = \frac{\tilde{w}_n}{\sum_{j=1}^N \tilde{w}_j} $$

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(c) KL-Implied Historical Weights (Weighted Monte Carlo)

Minimize KL divergence to a prior $\hat{p}_n$ subject to calibration constraints:

$$ \min_{p \in \Delta_N} \sum_{n=1}^N p_n \log \frac{p_n}{\hat{p}_n} $$

subject to:

$$ \sum_{n=1}^N g_{m,n} p_n = 0, \quad m = 1, \dots, M $$

where $g_{m,n}$ = delta-hedged P&L of instrument $m$ on path $n$ under market implied volatility.

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Use Cases

• Compare BEVL vs IV → detect volatility risk premium
• Fair skew estimation → construct “virtual” skew curve without IV noise
• Risk management → Gamma-weighted realized variance reflects true hedging risk
• Research → methodology for linking realized & implied volatility

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Example

python examples/run_example.py --csv data/SPX_sample.csv --tenor_days 21 --strikes 0.9 1.0 1.1